Formal Inverse Integrating Factor and the Nilpotent Center Problem

We are interested in deepening the knowledge of methods based on formal power series applied to the nilpotent center problem of planar local analytic monodromic vector fields 𝒳. As formal integrability is not enough to characterize such a center we use a more general object, namely, formal inverse integrating factors V of 𝒳. Although by the existence of V it is not possible to describe all nilpotent centers strata, we simplify, improve and also extend previous results on the relationship between these concepts. We use in the performed analysis the so-called Andreev number n ∈ ℕ with n ≥ 2 associated to 𝒳 which is invariant under orbital equivalency of 𝒳. Besides the leading terms in the (1,n)-quasihomogeneous expansions that V can have, we also prove the following: (i) If n is even and there exists V then 𝒳 has a center; (ii) if n = 2, the existence of V characterizes all the centers; (iii) if there is a V with minimum “vanishing multiplicity” at the singularity then, generically, 𝒳 has a center.

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